answer:To solve for z in the equation zleft(1+iright)=2-2i, we first isolate z by dividing both sides of the equation by 1+i: z = frac{2-2i}{1+i}. Next, to simplify the fraction, we multiply both the numerator and the denominator by the conjugate of the denominator, which in this case is 1-i: z = frac{(2-2i)(1-i)}{(1+i)(1-i)}. Expanding both the numerator and the denominator: z = frac{2(1-i)-2i(1-i)}{1^2 - i^2}. Since i^2 = -1, we simplify further: z = frac{2-2i-2i+2i^2}{1 - (-1)} = frac{2-4i-2}{2}. Substituting i^2 = -1 into the equation: z = frac{2-4i-2(-1)}{2} = frac{2-4i+2}{2} = frac{4-4i}{2} = 2-2i. However, this seems to be a misstep in the simplification process. Let's correct it from the multiplication step: z = frac{(2-2i)(1-i)}{1+1} = frac{2-2i-2i+2i^2}{2} = frac{2-4i-2}{2} = frac{-2i}{2} = -i. This indicates that the correct simplification should directly follow from multiplying the numerator and denominator by the conjugate, without the misstep: z = frac{(2-2i)(1-i)}{(1+i)(1-i)} = frac{2-2i-2i+2}{2} = 1-i. Upon realizing the mistake, let's accurately follow the original solution's correct path: z = frac{(2-2i)(1-i)}{(1+i)(1-i)} = frac{2-2i-2i+2i^2}{1 - i^2} = frac{2-4i-2}{2} = 1-2i. The correct simplification from the original solution is: z = frac{(2-2i)(1-i)}{(1+i)(1-i)} = left(1-iright)^{2} = -2i. Thus, the imaginary part of z is -2. Therefore, the correct choice is: boxed{D}.

answer:Given: The circumference of a circle is (100 pi text{ cm}). We know that the formula for the circumference (C) of a circle in terms of its radius (r) is: [ C = 2 pi r ] 1. Substitute the given circumference into the formula: [ 2 pi r = 100 pi ] 2. To isolate (r), divide both sides of the equation by (2 pi): [ r = frac{100 pi}{2 pi} ] 3. Simplify the fraction by canceling out (pi): [ r = frac{100}{2} ] 4. Calculate the result: [ r = 50 text{ cm} ] Conclusion: Therefore, the radius of the circle is (boxed{50 text{~cm} (C)}).

answer:The interviewees consist of 6 students, 4 junior high students, 2 teachers, 5 table tennis enthusiasts, and 2 basketball enthusiasts. - Since table tennis and basketball enthusiasts could be either students or teachers: - **Minimum number**: Assume the maximum overlap between the categories. Therefore, all junior high students are included in the count of students, and all the enthusiasts could also be students or teachers. So, the minimum number is just the count of students and teachers, which is 6 + 2 = 8. [boxed{8 text{ people}}] - **Maximum number**: Assume no overlap except the junior high students being a part of the broader student category. Additionally, assuming that each enthusiast is a unique individual who is neither a student nor a teacher, and that one cannot be both a table tennis enthusiast and a basketball enthusiast simultaneously: - Add the number of students not including junior high students (6 - 4 = 2 non-junior high students), the number of teachers, and both types of enthusiasts: 2 + 2 + 5 + 2 = 11. - Then add the junior high students, since they've been deducted from the student count: 11 + 4 = 15. [boxed{15 text{ people}}]

answer:The probability P(1 le x le 3) corresponds to the probability that the random variable falls within one standard deviation from the mean (which is 2) in a standard normal distribution, because the standard deviation is 1. In a standard normal distribution, the probability within one standard deviation from the mean (from -1 to 1) is approximately 0.6826 or 68.26%. The distribution is symmetric about the mean, so the probability on either side of the mean within one standard deviation is half of 0.6826, that is 0.3413. Therefore, the probability that the random variable falls below one standard deviation from the mean (to the left of the mean) is also 0.3413. This includes both P(1 le x le 2) and P(x < 1). Since we are symmetrically distributed about the mean, P(1 le x le 2) = 0.3413. This means that we must subtract this probability from 0.5 (the total probability to the left of the mean) to find P(x < 1). So, P(x < 1) = 0.5 - P(1 le x le 2) = 0.5 - 0.3413 = 0.1587. Hence, the answer is boxed{B: 0.1587}.